Forum:Pop the balloons ( googology challenge )
Capital letters are balloons, except for P which is a pin, and S which is the score, it begins at 0 At each step, gently push the balloons to the left, so whichever balloon is in contact with the pin "pops" or disappears, what happens next depends on the color of the balloon Red ballons R = Increase S by 1 Blue balloons B = Spawns S Red Balloons where they pop Green balloons G = Spawns S Blue Balloons where they pop Yellow balloons Y = Spawns S Green Balloons where they pop Note: Spawning ballons will push existing balloons to the right, if there are any, to make enough room for themselves Example run: S=0 PRRRG S=1 PRRG S=2 PRG S=3 PG S=3 PBBB S=3 PRRRBB S=4 PRRBB S=5 PRBB S=6 PBB S=6 PRRRRRRB ... (6 steps later) S=12 PB S=12 PRRRRRRRRRRRR ... (12 steps later) S=24 P The challenge is this: how much score would you gain from clearing the following wave of ballons? PRBGY If you give up scroll down for the answer Filler text for scrolling Filler text for scrolling Filler text for scrolling Filler text for scrolling Filler text for scrolling Filler text for scrolling Filler text for scrolling Filler text for scrolling Filler text for scrolling Filler text for scrolling Filler text for scrolling Filler text for scrolling Filler text for scrolling Filler text for scrolling Filler text for scrolling Filler text for scrolling Filler text for scrolling Filler text for scrolling Filler text for scrolling Filler text for scrolling Filler text for scrolling Filler text for scrolling Filler text for scrolling Filler text for scrolling Filler text for scrolling Filler text for scrolling Filler text for scrolling Filler text for scrolling Filler text for scrolling Filler text for scrolling Filler text for scrolling Filler text for scrolling Filler text for scrolling Filler text for scrolling Filler text for scrolling Filler text for scrolling Filler text for scrolling Filler text for scrolling Filler text for scrolling Filler text for scrolling Filler text for scrolling Filler text for scrolling Filler text for scrolling Filler text for scrolling Filler text for scrolling Filler text for scrolling Filler text for scrolling Filler text for scrolling Filler text for scrolling Filler text for scrolling Filler text for scrolling Filler text for scrolling Filler text for scrolling Filler text for scrolling Filler text for scrolling Filler text for scrolling Filler text for scrolling Filler text for scrolling Filler text for scrolling Filler text for scrolling Filler text for scrolling Filler text for scrolling Filler text for scrolling Filler text for scrolling Filler text for scrolling Filler text for scrolling Filler text for scrolling Filler text for scrolling Filler text for scrolling Filler text for scrolling Filler text for scrolling Filler text for scrolling Filler text for scrolling Filler text for scrolling Its, in fgh: f_3(8) Or, bound using tetration ^{10}2 < \ S < \ ^{11}2 : Cool idea. Just a minor correction: f3(8) is between 2↑↑10 and 2↑↑11: : 2↑↑10 ~ E4.3#7 : f3(8) ~ E619#7 : 2↑↑11 ~ E19728#7 PsiCubed2 (talk) ... corrected!, thanks psi Chronolegends (talk) 16:23, February 15, 2017 (UTC)12:14, February 15, 2017 (UTC) How was it originally?Boboris02 (talk) 18:12, February 25, 2017 (UTC) @boboris: i mistakenly bound at 2↑↑9 < f3(8) < 2↑↑10Chronolegends (talk) 18:35, February 25, 2017 (UTC) :I will give the explanation why that wave have the score \(f\_3(8)\). :First, there's a pin in the first one. :Each step, you have to pop the balloon after the pin. :*If there's red balloon after the pin then it increases the score by 1. \(\alpha \rightarrow f_0(\alpha)\) :*If there's blue balloon after the pin then it doubles the score. \(\alpha \rightarrow f_1(\alpha)\) :*If there's green balloon after the pin then it doubles the score, repeating depends how many score you get. \(\alpha \rightarrow f_2(\alpha)\) :*If there's yellow balloon after the pin then it doubles the score, 'repeating depends how many score you get' twice. \(\alpha \rightarrow f_3(\alpha)\) :We start at 0. :First we pop the red balloon, to set the score by 1. :Then we pop the blue balloon, to set the score by 2 (by doubling 1). :Then we pop the green balloon, to set the score by 8 (by doubling 2 twice). :Then we pop the yellow balloon, to set the score by \(f_3(8)\). Googleaarex (talk) 18:50, February 25, 2017 (UTC)